Consistency of the estimator of cumulative hazard function in estimated pseudo-partial-likelihood approach

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For multivariate failure time with auxiliary covariate information, an estimated pseudo-partial-likelihood estimator under the marginal hazard model with distinguishable baseline hazard has been proposed. However, the asymptotic properties of the corresponding estimated cumulative hazard function have not been studied. In this paper, based on counting process martingale, we use the continuous mapping theorem and Lenglart inequality and prove the consistency of the estimated cumulative hazard function in estimated pseudo-partial-likelihood approach.